Engineering Studio

Topic: Optimization (Volume)
Engineer:
Date:
📦 Project 1: The Rectangular Delivery Box
Design a box from a Rectangular sheet (15 cm by 8 cm). Cut squares of side \( x \) from each corner.
x 15 cm 8 cm
1. Constraints

The shortest side is 8 cm. If we cut \( x \) from both ends, what is the max limit for \( x \)?

Constraint: \( 0 < x < \) ______ (cm)
2. Dimensions
Height = \( x \)
Length = \( 15 - ... \)
Width = \( 8 - ... \)
3. The Model

Show that Volume \( V(x) = 4x^3 - 46x^2 + 120x \).

4. Optimization (Calculus)

a) Find \( \frac{dV}{dx} \):

b) Solve \( \frac{dV}{dx} = 0 \).

5. Decision Making
Value 1: \( x = \) ______
[ ] Accept
[ ] Reject
Value 2: \( x = \) ______
[ ] Accept
[ ] Reject

Max Volume: _______________ cm³

Land Management

Topic: Optimization (Area)
🏡 Project 2: The Three-Sided Fence
Build a rectangular enclosure using 20m of fencing against a wall. Maximize Area.
Existing Stone Wall x x y AREA
1. Variables & Constraint

Total orange fence = 20m. Width = \( x \), Length = \( y \).

Constraint (Perimeter): \( ... + ... = ... \)  →  \( y= \) ______________
2. Objective Function

Maximize Area \( A = x \times y \). Substitute \( y \) to get \( A(x) \):

\( A(x) = \)
3. Optimization

Find \( \frac{dA}{dx} \) and solve for zero.

4. Final Design
Width \( x = \) ______ m
Length \( y = \) ______ m

Max Area: _______ m²

🧠 Challenge: The Cost Factor

Scenario: The fence parallel to the wall (\( y \)) is stronger and costs $10/m. The side fences (\( x \)) cost $5/m. Total Budget = $200.

Equation: \( 5(2x) + 10(y) = 200 \).

Question: Without calculus, use your intuition: To maximize area, should we make the expensive side (\( y \)) shorter or longer?